Sophie: I haven't carefully checked those bag cost and coupon discount examples, so I can't promise that \\(f(x) \otimes_Y f(x') = f(x \otimes_X x') \\) is not necessary. But I'm willing to believe you.

Re-examining what I wrote, it seems that a necessary and sufficient condition is

\[ f(x_1)\otimes_Y f(x_2) \le_Y f(x_1') \otimes_Y f(x_2') \textrm{ implies } f(x_1 \otimes_X x_2) \le_Y f(x_1' \otimes_X x_2') .\]

It's late, so I'll have to check this when I'm more awake. Does this condition hold in your examples?

Re-examining what I wrote, it seems that a necessary and sufficient condition is

\[ f(x_1)\otimes_Y f(x_2) \le_Y f(x_1') \otimes_Y f(x_2') \textrm{ implies } f(x_1 \otimes_X x_2) \le_Y f(x_1' \otimes_X x_2') .\]

It's late, so I'll have to check this when I'm more awake. Does this condition hold in your examples?